Mathematics Subject GRE
The Mathematics GRE consists of about 70 questions. 50% of the exam is calculus and its applications. 25% consists of elementary algebra, linear algebra, abstract algebra and number theory. The remaining 25% consists of others areas commonly studied by undergraduates. Therefore, a thorough review of calculus is very important.
The GRE Subject Mathematics site includes a practice booklet. Based on that, it appears that the most important classes to take beyond Math 11 - 14, 22, 52 and 53 are (in descending order of importance) Math 153, 154, 111, 105, 103, 122. There were single questions from Math 113, 175, 177.
For additional preparation, you can purchase Princeton Review's Cracking the GRE Mathematics Subject Test, 4th Edition.
The calculus questions cover pretty much everything we teach in Math 11 - 14. Here is a very small sample of things that appear on the exam from these courses:
- interval of convergence of a power series,
- Green's theorem,
- a continuous function on a closed, bounded set takes on maximum and minimum values,
- if f is continuous on [b,c] then there's a d in [b,c] such that the integral from b to c of f equals f(d)(c-b).
Much, though not all, of the 25% algebra is covered in our Math 52 and 53. An important source of review would be the theorems in Math 52 about finite groups and their subgroups. From Math 53, know about matrices and operations on them, vector spaces and their subspaces, eigenvalues and characteristic polynomials.
The remaining 25% of the exam includes topics from elementary topology of R (the reals) and R^n, properties of continuous functions, differentiability and integrability, general topology, complex variables, probability and statistics, set theory, logic, combinatorics, discrete mathematics, algorithms and numerical analysis. There were also some problems about tracing through an algorithm and describing the output; a student who completed CSCI 10 should have no problem with these.